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Math Help - minkowski inequality

  1. #1
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    minkowski inequality

    Let (X,F,\mu)=([0,a],\beta, \Lambda) be the usual Lebesgue measurable space.
    Using Holder inequality, prove Minkowski inequality for any f \in L^2(X),g \in L^2(X), that is ||f+g||_2 \leq ||f||_2 +||g||_2

    i am stuck and dont know what to do. any help would be appreciated.
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  2. #2
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    Imitate the proof of the triangle inequality. (that is start by squaring both sides) Then you can use Holder's inequality with p=q=2
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  3. #3
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    Quote Originally Posted by putnam120 View Post
    Imitate the proof of the triangle inequality. (that is start by squaring both sides) Then you can use Holder's inequality with p=q=2
    i am not sure what i should square the both sides of. holders inequality?
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  4. #4
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    \parallel f+g\parallel_2^2=\int (f+g)^2=\int f^2+\int g^2+2\int fg

    (\parallel f\parallel_2 +\parallel g\parallel_2)^2=\int f^2+\int g^2+2\parallel f\parallel_2\parallel g\parallel_2

    From here you should be able to see how to use Holder's.
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